Teleparallel dark energy · 2013-12-06 · Teleparallel dark energy Chung-Chi Lee Teleparallel...

Teleparalleldark energy

Chung-ChiLee

TeleparallelGravity

TeleparallelDark Energy

ObservationalConstraints

TrackerBehavior

Teleparallel dark energy

Chung-Chi Lee

National Tsing Hua University, Taiwan

May. 10, 2012

Based on:CQ Geng, CC Lee, E. N. Saridakis, YP Wu PLB 704, 384 (2011)CQ Geng, CC Lee, E. N. Saridakis JCAP 1201, 002 (2012)

JA Gu, CC Lee, CQ Geng arXiv:1204.4048


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Outline

Teleparallel Gravity

Teleparallel Dark Energy model

Observational Constraints

Tracker Behavior


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Teleparallel GravityWhat is the feature of teleparallel gravity?

An alternative theory of gravity, which is equivalent toGeneral Relativity.

This is a curvatureless gravity theory, and the gravitationaleffect comes from torsion instead of curvature.


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Teleparallel GravityA brief introduction

The dynamical variable of teleparallel gravity is thevierbein fields eA(x

μ), which form an orthonormal basisfor the tangent space at each point xμ of the manifold:eA · eB = ηAB , where ηAB = diag(1,−1,−1,−1).

Notation:Greek indices μ, ν,... : coordinate space-time.Latin indices A,B,... : tangent space-time.

The relationship between metric and vierbein fields is

gμν(x) = ηAB eAμ (x) eBν (x).


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Weitzenbock connection: a curvatureless connection

wΓλ

νμ ≡ eλA ∂μeAν

The torsion tensor is defined as

T λμν ≡

wΓλ

νμ −wΓλ

μν = eλA (∂μeAν − ∂νe

Aμ ).

Under Weitzenbock connection, the Riemann tensorvanishes:

Rρμσν =

wΓρ

μν,σ −wΓρ

μσ,ν +wΓρ

δσ

wΓδ

μν −wΓρ

δν

wΓδ

μσ = 0.


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We can construct the “teleparallel Lagrangian” by usingthe torsion tensor,LT = T = a1T

ρμνTρμν + a2TρμνTνμρ + a3T

ρρμ T μν

ν .

It is a good approach of General Relativity when we choosethe suitable parameters a1 =

14 , a2 =

12 and a3 = −1:

R = −T − 2∇μT νμν ,

where R is constructed by Levi-Civita connection.


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The action of teleparallel gravity is

S =

∫d4xe

[T

2κ2+ LM

],

where e = det(eAμ

)=

√−g.

Varying this action respect to the vierbein fields gives thefield equation

e−1∂μ(eeρASρ

μν)− eλATρμλSρ

νμ − 1

4eνAT =

κ2

2eρA

emT ρ

ν ,

whereemT ρ

ν stands for the energy-momentum tensor andSρ

μν = 14 (T

νμρ − T μν

ρ + T μνρ ) + 1

2

(δμρ Tαν

α − δνρ Tαμα

).


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Teleparallel Dark EnergyWhat is teleparallel dark energy model?

Teleparallel dark energy model is a dark energy model,which can explain the late time accelerating universe.

This model combines quintessence model with teleparallelgravity.

This model differs from quintessence model when we turnon the non-minimal coupling term.


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Teleparallel Dark EnergyA brief review of quintessence model

Quintessence is one of the most popular dark energymodel.

The generalized quintessence model action is given by

S =

∫d4x

√−g

[R

2κ2+

1

2

(∂μφ∂

μφ+ ξRφ2)− V (φ) + LM

]

Under the flat Friedmann-Robertson-Walker (FRW)background ds2 = dt2 − a2(t)dx2, the effective energy andpressure density can be defined as

ρφ =1

2φ2 + V (φ) + 6ξHφφ+ 3ξH2φ2,

pφ =1

2φ2 − V (φ) + ξ

(2H + 3H2

)φ2 + 4ξHφφ

+2ξφφ+ 2ξφ2


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Teleparallel Dark Energy

Similar to quintessence model, we can constructteleparallel dark energy model, and the action is given by

S =

∫d4xe

[T

2κ2+

1

2

(∂μφ∂

μφ+ ξTφ2)− V (φ) + LM

].

Variation of action with respect to the vierbein fields yieldsthe field equation(

2

κ2+ 2ξφ2

)[e−1∂μ(ee

ρASρ

μν)− eλATρμλSρ

νμ − 1

4eνAT

]

−eνA

[1

2∂μφ∂

μφ− V (φ)

]+ eμA∂

νφ∂μφ

+4ξeρASρμνφ (∂μφ) = eρA

emT ρ

ν .


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Again, the effective energy and pressure density underFRW metric (eAμ = diag(1, a, a, a)) are

ρφ =1

2φ2 + V (φ)− 3ξH2φ2,

pφ =1

2φ2 − V (φ) + 4ξHφφ+ ξ

(3H2 + 2H

)φ2.

Variation of action with respect to the scalar field gives usthe equation of motion of the scalar field

φ+ 3Hφ+ 6ξH2φ+ V ′(φ) = 0.

These equations lead to the continuity equationρφ + 3H(1 + wφ)ρφ = 0, where wφ is the equation ofstate of the scalar field, which is defined as wφ ≡ pφ

ρφ.


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In the minimal coupling case (ξ = 0), the teleparallel darkenergy is equivalent to quintessence model

However, these two models are different theories when weturn on the non-minimal coupling constant (ξ �= 0)

Teleparallel dark energy model can cross thephantom-divide easily.

Similar to f(T ) theory,this model has the localLorentz violation problem.


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Teleparallel Dark Energy:Observational Constraints

We would like to test teleparallel dark energy model byusing the SNIa, BAO and CMB data. These observationaldata can tell us whether this is a suitable model for darkenergy or not

We consider three kinds of potential cases:Power-Law potential: V (φ) = V0φ

4

Exponential potential: V (φ) = V0e−κφ

Inverse hyperbolic cosine potential: V (φ) = V0cosh(κφ)


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Potential: V (φ) = V0φ4

Left: fixing Ωm = 27%, the best fit locates at h � 0.7,ξ � −0.42, wφ � −0.96 and χ2 � 543.9

Right: fixing ξ = −0.41, the best fit locates at h � 0.7,Ωm � 28.0%, wφ � −0.99 and χ2 � 544.5


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Potential: V (φ) = V0e−κφ




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Potential: V (φ) = V0cosh(κφ)




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Summary

Teleparallel gravity is an alternative gravity theory of theuniverse.

Teleparallel dark energy model is equivalent toquintessence model happens at the minimal coupling case(ξ = 0), but it has a different behavior when we include anon-minimal coupling term (ξ �= 0).

We show that the equation of state of teleparallel darkenergy model can cross the phantom-divide easily.

The observational constraints show a good result on thismodel. This model is suitable for the late-timeaccelerating universe.


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Tracker BehaviorBasic Idea and Features

Potential-free: V (φ) = 0.

Analytic solutions in the radiation (RD), matter (MD),and scalar field (SD) dominated eras.

The tracker behavior for wφ in the RD and MD eras.


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Tracker BehaviorBasic Idea and Features

The field equation of gravity and scalar field lead to

φ+ 3Hφ+ 6ξH2φ = 0 ,

H2 ≡(a

a

)2

=κ2

3(ρφ + ρm + ρr) ,

H = −κ2

2(ρφ + pφ + ρm + 4ρr/3) .

The effective energy and pressure density are

ρφ =1

2φ2 − 3ξH2φ2 ,

pφ =1

2φ2 + 3ξH2φ2 + 2ξ

d

dt(Hφ2) .


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Tracker BehaviorAnalytic Solutions in RD and MD eras

H = α/t, i.e. a(t) ∝ tα, with α constant:

φ(t) = C1tl1 + C2t

l2 ,

where C1,2 are constants and

l1,2 =1

2

[±√

(3α− 1)2 − 24ξα2 − (3α− 1)].

For ξ < 0, the power-index l1 is positive and l2 is negative,corresponding to increasing and decreasing modes,respectively.

Considering only the increasing mode, i.e., φ(t) = C1tl1 .


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Tracker BehaviorTracker Behavior in RD and MD eras

We can solve the analytic solution in RD and MD eras:

wφ =1

3

(2−

√1− 24ξ

),

1

2

(1−

√1− 32ξ/3

),

ρφ ∝ a−5+√1−24ξ , a(−9+

√9−96ξ )/2,

for the RD (α = 1/2) and MD (α = 2/3) eras,respectively.


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Tracker BehaviorAnalytic Solutions in SD era

Setting ρr = 0:

d

dt

[F (φ)a3H

]=

κ2

2ρma

3 =3

2H2

0Ωm0 ,

F (φ) ≡ 1 + κ2ξφ2.

Setting ρm = 0 and combining with field equation, we cansolve the analytic solution:

φ(a) = ± sin θ√−κ2ξ

,

wφ(a) = −1−√

−32ξ/3 tan θ ,

θ(a) ≡√

−6ξ ln a+ C3 ,

where C3 is the integration constant.There exists one kind of finite-time future singularities,called “sudden singularity”.


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Tracker BehaviorNumerical Result

The tracker behavior in RD (MD) eras,and the singularity happens in SD era.

The present (z = 0) values of (Ωm, wφ)with ξ = −0.35.


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Tracker BehaviorNumerical Result

Combining SNIa, BAO and CMB data:


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Summary

Analytic solution exist in this potential-free case.

Equation of state (wφ) has a tracker behavior and onlydepend on ξ in RD and MD eras.

The final energy density depends on ξ and initial conditionparameter C1.

The observational data can be fitted well but theconcordance region for all data is only at the 3σ level.

Teleparallel dark energy · 2013-12-06 · Teleparallel dark energy Chung-Chi Lee Teleparallel...

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